Scheduling n jobs on one machine to minimize the maximum tardiness with minimum number

The “one machine” scheduling problem is considered with the dual objective of minimizing the maximum tardiness with minimum number of tardy jobs. A simple procedure is introduced to obtain an optimal schedule with minimum “maximum tardiness” when the set of nontardy jobs is specified. A branch and bound algorithm is presented to obtain the optimal schedule that minimizes the maximum tardiness with minimum number of tardy jobs. A condition is also given to identify an initial set of early jobs. Several theorems are formulated and proved in order to justify the elimination of much branching.